Constructing a Triangle in Higher Dimensions

[roger]

hi

In an earlier thread about ''lengths that are irrational'', matt Grime said that if we were to construct a triangle etc

What I wanted to ask was is it possible to construct a 2 dimensional shape in real life ? even though we live in a 4 dimensional universe ?

Thanks

Roger

 

[mathman]

You can draw a two dimensional shape on a piece of paper. However, if you mean actually build something out of stuff, the answer is no. All stuff has three (more in string theory) dimensions.

 

[roger]

You can draw a two dimensional shape on a piece of paper. However, if you mean actually build something out of stuff, the answer is no. All stuff has three (more in string theory) dimensions.

But strictly speaking, is the drawing of the shape really 2 dimensional ??
because I heard that it can only be seen as light is reflected off the pencil drawing which has a infinitesimally small thickness ?
Thats why I wanted to ask whether the shape can exist or if its just in the mind ?

Also, why does an electron have zero dimensions, and yet can exist in three dimensions ?

thanks for any info

Roger

 

[Gokul43201]

No, it's not strictly two dimensional, but the thickness (or depth of penetration, or whatever it is that needs to be at least an atom thick) is small compared the other dimensions. So, for all practical purposes, it's is fair to call it 2D.

 

[Gokul43201]

In mathematics, points, lines and planes exists in 3D space, so why can you not have the equivalent of a point in reality. An alternative (hand waving) argument is that you can imagine the electron as a (three dimensional) sphere with zero radius.

A full explanation would be more complex.

 

[dextercioby]

But strictly speaking, is the drawing of the shape really 2 dimensional ??

According to geometry,yes.

because I heard that it can only be seen as light is reflected off the pencil drawing which has a infinitesimally small thickness ?
Thats why I wanted to ask whether the shape can exist or if its just in the mind ?

Don't mix geometry (and its concepts) with physics (optics).Not in this case, actually,as geometry is a useful/essential tool in optics.But in this case,the triangle is a mathematical abstraction,it has no other dimensions except for the lines/sides seen as segments of a line,where the last notion is understood geometrically.No thickness,no depth,no width,just lenght.It's something abstract and idealized.Mathematics uses commonly such concepts.

Also, why does an electron have zero dimensions, and yet can exist in three dimensions ?

Again you're mixing physics with geometry.Wrongly,that is.A point is a notion of geometry.A point can exist in any dimension space possible,since it has zero dimension.By the way,the notion of "space dimension" should belong to differential geometry,where it's stated as "manifold dimension".
An electron is a pointlike particle,that is a particle that has no space dimentions (irrelevant of number of the space (space-time) dimentions the space that contains it has).Why...?It's considered as a fundamental particle (that is,no composite/internal strucure) and because theories that describe it (QM and QFT) by definition consider electron as a mere point in space time,to which we attach some numbers with physicsl segnificance.
To conclude,"points" are merely geometrical abstraction with no physical relevance/existence whatsoever.Yet,most of fundamental phyiscs has been built on the assumption that (fundamental) particles are "pointlike" .

BTW,there have been made calculations on the hydrogen atom in which the proton/nucleus was assumed finite size.And not because it's not fundamental anymore,as it's filled with quarcks and gluons,because we MUST see those particles as finite size,as they actually are.

The idea of QM pointlike particle physics is completely rejected by string theory.

 

[roger]

According to geometry,yes.



Don't mix geometry (and its concepts) with physics (optics).Not in this case, actually,as geometry is a useful/essential tool in optics.But in this case,the triangle is a mathematical abstraction,it has no other dimensions except for the lines/sides seen as segments of a line,where the last notion is understood geometrically.No thickness,no depth,no width,just lenght.It's something abstract and idealized.Mathematics uses commonly such concepts.



Again you're mixing physics with geometry.Wrongly,that is.A point is a notion of geometry.A point can exist in any dimension space possible,since it has zero dimension.By the way,the notion of "space dimension" should belong to differential geometry,where it's stated as "manifold dimension". please can someone explain this bit ?
An electron is a pointlike particle,that is a particle that has no space dimentions (irrelevant of number of the space (space-time) dimentions the space that contains it has).Why...?It's considered as a fundamental particle (that is,no composite/internal strucure) and because theories that describe it (QM and QFT) by definition consider electron as a mere point in space time,to which we attach some numbers with physicsl segnificance.
To conclude,"points" are merely geometrical abstraction with no physical relevance/existence whatsoever.Yet,most of fundamental phyiscs has been built on the assumption that (fundamental) particles are "pointlike" .

BTW,there have been made calculations on the hydrogen atom in which the proton/nucleus was assumed finite size.And not because it's not fundamental anymore,as it's filled with quarcks and gluons,because we MUST see those particles as finite size,as they actually are.

The idea of QM pointlike particle physics is completely rejected by string theory.

so if string theory is proven, will qm theory be wrong ?
I read somewhere that the point is the end of the string ? Is this true ?


But if a point is zero dimensional as stated above, how can a series of points make up a line ?
likewise, how can a line make up an area if the line has no width or height ?

thanks

roger.

 

[roger]

Please would somebody else care to respond to my queries above.

Thankyou

Roger

 

[matt grime]

The response is, as ever it appears, that you are confusing a mathematical model with the phyisical objects and phenomena involved.

QM, say, is correct in that it accurately models observed phenomena. It may be superseded by a better model that applies in greater generality but that doesn't make it wrong per se, any more than Newton's Laws are wrong: they just don't apply in some cases.

 

[roger]

The response is, as ever it appears, that you are confusing a mathematical model with the phyisical objects and phenomena involved.

QM, say, is correct in that it accurately models observed phenomena. It may be superseded by a better model that applies in greater generality but that doesn't make it wrong per se, any more than Newton's Laws are wrong: they just don't apply in some cases.

And what did he mean by manifold dimension ?

 

[matt grime]

The dimension of a manifold is the dimension of the tangent bundle at all points (assuming it is a global constant).

 

[Rogerio]

...
Is it possible to construct a 2 dimensional shape in real life ?

Of course YES !

Use shadows !

:-)

 

[roger]

Of course YES !

Use shadows !

:-)

but what is a manifold in simple words ?

I'm still a high school student, I don't have a clue !


Roger

 

[matt grime]

An example of manifold is a subset of (and let's keep it visualizable) space that is "locally" like the plane. For instance the surface of a sphere is locally 2-d - that means around a point there's a little patch that looks like a bit of the plane.

If you think topographically, whenever we want to model a small portion of the Earth's surface we use a flat map around that point. If we pick to different maps that overlap then they should agree on the overlap. A collection of maps is called an atlas. All these ideas can be put together formally and the result is called a manifold. It means that we can use properties of the plane (in this case) to reason "locally", and how these local maps glue together to reason "globally".